where the integration takes over the homology \(\delta \)  between \(Z_{s_0}\)  and \(Z\) : \(\partial \delta =Z_s-Z_{s_0}\) . For a proof of the existence of such a homology path see [1]}, however, note that from this proof it is not clear whether \(\delta \)  is supported in \(\cup _{s\in {\sf S }}Z_s\)  or not. If the parameter space \({\sf S }=\mathbb {P}^1\)  is the rational curve, it is well-known that there is no non constant holomorphic map from \(\mathbb {P}^1\)  to a complex compact tori, and hence, the Abel-Jacobi map is zero in this case. This implies that for some homology \(\delta \)  between \(Z_{s_0}\)  and \(Z_s\)  we have
in \(\int _{\delta }F^2=0\) , but not necessarily \(\delta \)  is supported in \(\cup _{s\in \mathbb {P}^1}Z_s\) .
There are examples of this situation for which we have this stronger statement.
